Momentum operator rotation

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

Again, the rotational kinetic energy, which is the full rotational Hamiltonian, can be written in terms of the total rotational angular momentum operator J2 and the component of angular momentum along the axis with the unique principal moment of inertia ... 

Again, the square of the total rotational angular momentum operator appears in Hj-ot... 

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

ANGULAR MOMENTUM OPERATORS AND ROTATIONS IN SPACE AND TRANSFORMATION THEORY OF QUANTUM MECHANICS ... 

We first inquire as to the constants of the motion in this situation. Since h is invariant under the group of spatial rotations, and under spatial inversions, the total angular momentum and the parity operator are constants of the motion. The total angular momentum operator is... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

With these results for the angular-momentum operators it is possible to obtain die Hamiltonian for the rotation of a symmetric top by direct substitution in Eq. (13). The leader is warned that care must be taken in this substitution, as die order of the derivatives is to be rigorously respected. However, given sufficient patience one can show that the classical energy becomes the Hamiltonian operator in the form (problem 12)... 

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

Because of the spherical symmetry of physical space, any realistic physical operator (such as the Schrodinger operator) must commute with the angular momentum operators. In other words, for any g e SO(3) and any f in the domain of the Schrodinger operator H we must have H o p(g ] = pig) o H, where p denotes the natural representation of 80(3 on L2(] 3 Exercise 8.15 we invite the reader to check that H does indeed commute with rotation. The commutation of H and the angular momentum operators is the infinitesimal version of the commutation with rotation i.e., we can obtain the former by differentiating the latter. More explicitly, we differentiate the equation... 

The spherical-top Hamiltonian is Htot= P2/1I. The ellipsoid of inertia is a sphere. The spherical-top rotational wave functions can thus be classified according to the symmetry species of %h. (The angular-momentum operator P2 =P2 + Pf+P2 remains unchanged no matter how the abc axes are rotated.)... 

Components of the angular momentum operator are connected with the infinitesimal operators of the group of rotations in three-dimensional space [11]. 

Equations (11.3.23) and (20) show that the infinitesimal generator I of rotations in ft3 about any axis n is the angular momentum about n. The separate symbol I has now served its purpose and will henceforth be replaced by the usual symbol for the angular momentum operator, J, and similarly /), I2, h will be replaced by Jx, Jy, Jz. 

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